F3_Individual Income Taxation

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Chapter 3 Individual Income Taxation

3.1 Introduction

The individual income tax is the most important single tax in many countries. The basic

principle of the individual income tax is that the taxpayers income from all sources should be

combined into a single or global measure of income. Total income is then reduced by certain

exemptions and deductions to arrive at income subject to tax. This is the base to which tax rate

are applied when computing tax.

A degree and coverage of exemptions and deductions vary from country to country. A

degree of progressivity of tax rates also varies.

Nevertheless the underlying principles of the tax system are common among countries and

are worth reviewing.

3.2 The Income-Based Principle

Economists have argued that a comprehensive definition of income must be used that includes

not only cash income but capital gains. A number of other adjustments have to be made to

convert your cash income into the comprehensive income that, in principle, should form the

basis of taxation.

This comprehensive definition of income is referred to as the Hicksian concept or the

Haig-Simons concept. This concept measures most accurately reflects ability to pay.

(1)Cash basis: In practice, only cash-basis market transactions are taxed. The tax is thuslevied on a notion of income that is somewhat narrower than that which most economists

would argue. Certain non-marketed (non-cash) economic activities are excluded,

though identical activities in the market are subject to taxation (e.g. housewifes work at

home (vis--vis a maids work), and own house (vis--vis rented house)).

Some non-cash transactions are listed in the tax code but are difficult to enforce.

Barter arrangements are subject to tax.

Unrealized capital gains is also not included in the income tax bases. Capital gains

are taxed only when the asset is sold (not on an accrual basis).

(2)Equity-based adjustments: Individuals who have large medical expenses or casualtylosses are allowed to deduct a portion of those expenses from their income, presumably

Lectures on Public Finance Part2_Chap3, 2012 version P.1 of 43


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on the grounds that they are not in as good a position for paying taxes as someone with

the same income without those expenses.

(3)Incentive-based adjustments: The tax code is used to encourage certain activities byallowing tax credits or deductions for those expenditures. Incentives are provided for

energy conservation, for investment, and for charitable contributions.

(4)Special Treatment of Capital Income: The tax laws treat capital and wage incomeseparately. The difficulty of assessing the magnitude of the returns to capital plays some

role, while attempts to encourage savings as a source of domestic investment and growth.

The Progressivity Principle1

Even the simplification of tax schedule prevails among countries, the premise remains that

those with higher incomes not only should pay more but should pay a larger fraction of their

income in taxes. In other words, progressivity is reflected in an increase not only in average

rates but in marginal rates.

Defining progressive and regressive is not easy and, unfortunately, ambiguities in definition

sometimes confuse public debate. A natural way to define these words is in terms of the

average tax rate, the ratio of taxes paid to income. If the average tax rate increases with

income, the system is progressive; if it falls, the tax is regressive.

Confusion arises because some people think of progressiveness in terms of the marginal tax

rate the change in taxes paid with respect to a change in income. To illustrate the distinction,

consider the following very simple income tax structure. Each individual computes her tax bill

by subtracting $3,000 from income and paying an amount equal to 20 percent of the remainder.

(If the difference is negative, the individual gets a subsidy equal to 20 percent of the figure.)

Table 1 Tax Liabilities under a Hypothetical Tax System

($)

Income Tax Liability Average Tax Rate Marginal Tax Rate2,000 -200 -0.10 0.23,000 0 0 0.25,000 400 0.08 0.2

10,000 1,400 0.14 0.230,000 5,400 0.18 0.2

Table 1 shows the amount of tax paid, the average tax rate, and the marginal tax rate for each of

several income levels. The average rates increase with income. However, the marginal tax

rate is constant at 0.2 because for each additional dollar earned, the individual pays an

1 This Part draws heavily from Rosen (1999), pp.258-260.



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additional 20 cents, regardless of income level. People could disagree about the

progressiveness of this tax system and each be right according to their own definitions. It is

therefore very important to make the definition clear when using the terms regressive andprogressive. In the remainder of this section, we assume they are defined in terms of average

tax rates.

The degree of progression

Progression in the income tax schedule introduces disproportionality into the distribution of

the tax burden and exerts a redistributive effect on the distribution of income. In order to

explore these properties further, we need to be able to measure the degree of income tax

progression along the income scale. Such measures are called measures of structural

progression (sometimes, measures oflocal progression). There is more than one possibility, as

we will see. Each such measure will induce a partial ordering on the set of all possible income

tax schedules. We could not expect always to be able to rank a schedule

unambiguously more, or less, structurally progressive than another schedule : we must

allow that a schedule could display more progression in one income range and less in another.

)(2 xt

)(1 xt

)(xt

)(xm )(x

Nevertheless, the policymaker and tax practitioner, and indeed the man in the street, would

like to be able to say which of any two alternative income tax systems is the more progressive in

its effects. Is the federal income tax in the USA more redistributive than the personal income

tax in Germany? This sort of question will take us from measures of structural progression to

measures ofeffective progression. Measuring effective progression is a matter of reducing a

tax schedule and income distribution pair to a scalar index number. The same schedule

could be more progressive in effect when applied to distribution A than to distribution B.

Trends in effective progression for a given country over time, as well as differences between the

income taxes of different countries, can be examined using such index numbers.

Let us begin by defining as and a respectively the marginal and average rates of

tax experienced by an income x :

x

xtxa

)()( = and (1))(')( xtxm =

Since

x

xaxmx )()()( =

x

txxt

x

xxtd )('

d

/)]([2

= (2)



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for strict progression it is necessary and sufficient that

xxax allfor)()>

)(xm )(xa

m( (3)

The strict inequality rules out the possibility that the tax could be proportional to income in any

interval: we may relax it if we wish. Measures of structural progression quantify, in various

ways, the excess of the marginal rate over the average rate at income level x .

We introduce two particularly important measures here.

First, liability progression ) is defined at any income level(xLP x for which t as

the elasticity of tax liability to pre-tax income:

0)( >x

1)(

)(

)(

)('>=

xa

xm

xt

xxt)( ),( == exLP xxt (4)

As we have already noted, for a strictly progressive income tax a 1 percent increase in pre-tax

income x leads to an increase of more than 1 percent in tax liability. measures the

actual percentage increase experienced. A change of tax schedule which, for some , casues

an increase in connotes, in an obvious sense, an increase in progression at that income

level . If the change in a strictly positive income tax involves an upward shift of the entire

function , then the tax has become everywhere more liability progressive.

)(xLP

0x

)( 0xLP

0x

)(xLP

(xRPSecond, residual progression ) is defined at all income levels x as the elasticity of

post-tax income to pre-tax income:

1)(1

)(1

)

)]( 0, Ul < 0, Uxx < 0 (12)

and

U as l) l x l( , 1 (13)

(15) implies that each household will endeavour to avoid corner solutions with l=1 (no one

wants to work all day long!!). The indifference curves of the utility function are illustrated

bellows in which utility increases to the north west.



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Figure 3 Preference

I0I1

I2

l

x

To allow preferences and the budget constraint to be depicted on the same diagram, the

utility function can be written

U = U(x,l) = U(x, z/s) = u(x,z,s) (14)

The indifference curves ofu(x,z,s), drawn (z,x)-space are dependent upon the ability level of

the household since it takes a high-ability household less labor time to achieve any given level

of income.

In fact, the indifference curves are constructed from those in ( l,x)-space by multiplying by

the relevant value ofs. This construction for the single indifference curve I0 and households of

three different ability levels.

Figure 4 Translation of indifference curves.

x

l

I0

x

l

I0(S1)



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Agent Monotonicity

The utility function (14) satisfies agent monotonicity if uz/ ux , is a decreasing function ofs.

Note that

u

uz

xis the marginal rate of substitution between consumption and pre-tax

income and that agent monotonicity requires

ss

s1. Agent

monotonicity implies that any two indifference curves of households of different abilities only

cross once. In other words, the indifference curve of an s-ability individual through the point

(x,z)in consumption-labor space rotates strictly clockwise as s increases.

Figure 5

x

1s=ss=s2

z



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Mirrlees proved a theorem which shows, when the consumption function is a differentiable

function of labor supply, agent monotonicity implies that gross income is an increasing functionof ability (in other words, if agent monotonicity holds and the implemented tax function has

pre-tax income increasing with ability, then the second-order condition for utility maximization

must hold). This is important as to identify ones ability by watching gross income.

Self-selection

Letx(s) andz(s) represent the consumption and income levels that the government intends a

sehold of ability s to choose. The household of ability s will choose (x(s), z(s))provided

In case of linear taxation, it does not need to consider the self-selection constraints since the

can be determined as a function of the two parameters that describe

e tax function; the lump-sum payment and the marginal rate of tax.

tion, then the

self-selection constraint is satisfied. The idea is to induce the more able group to reveal that

th

o derive the required minimization problem, let u(s)=u(x(s), z(s), s) represent the maximized

(16)

hou

that this pair generates at least as much utility as any other choice. This condition must apply

to all consumption-income pairs and to all households. Formally we can write,

The self-selection constraint is satisfied if u(x(s), z(s), s) u(x(s), z(s), s) for all s ands.

behavior of the household

th

In case of non-linear taxation, the self-selection constraints must be included. This is

achieved by noting that the satisfaction of the self-selection constraint is equivalent to achieving

the minimum of a certain minimization problem. If the sufficient conditions for the

minimization are satisfied by the allocation resulting from the tax func

ey have a high income, not the reverse.

T

level of utility for a consumer of ability s resulting from (10).

0= u(s)-u(x(s), z(s), s) u(s)-u(x(s), z(s), s)

so that s =s minimizes u(s)-u(x(s), z(s), s). Hence

u(s)=us(x(s), z(s), s). (17)

From the definition ofu(s) it follows that



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uxx(s) + uzz (18)

Condit (17) or (18) is the necessary (the first order) condition for the self-selection

The second-order condition for the self-selection constraint is found from the second

(19)

szz(s) + uss (20)

.

(21)

(s) = 0

is equivalent to (17).ion

constraint to be satisfied.

derivative ofu(s)-u(x, z, s) with respect to s to be

u(s) - uss(x(s), z(s), s) 0

Again using the definition ofu(s),

u(s) usxx(s) + u

which gives, by using (19)

usxx(s) + uszz(s) 0

Eliminatingx(s) using (18) provides the final condition

u uu

z s z ssz sxz s

= ' ( ) ' ( )

u ux x 0 (22)

here s is the marginal rate of substitution introduced in the discussion of agent monotonicity.

s


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3.7 The General Problem

Using the individual demand and supply functions and integrating over the population, it is

nd aggregate demand,X, where

X x s s ds=

( ) ( )0 (24)

The optimal tax function is then chosen to maximize social welfare, where social welfare is

given by the Bergson-Samuelson function.

W w u s s ds

( ( )) ( )0 (25)

with W 0.

There are two constraints upon the maximization of (25). The first is that the chosen

allocation must be productively feasible such that,

XF(Z) (26)

where Fis the production function for the economy.

This definition of productive feasibility can incorporate the government revenue requirement,

expressed as a quantity of labor consumed by the government ZG, by noting that (26) can be

written )

Denoting the level of revenue required byR(ZG), the revenue constraint can be written

[ ]s x s s ds ( ) ( ) ( ) (27)

The second constraint is that it must satisfy the self-selection constraint which has already been

possible to define total effective labor supplyZ, by

Z z s s ds=

( ) ( )0 (23)

a

=

X F Z Z F ZG =$ ( ) ( .

R z

0



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discussed.

3.8 Linear taxation

With linear taxation the marginal rate of tax is constant and there is an identical lump-sum tax or

y for all househol

The advantages of this restriction is that it en

onvex so that optimal choices will be unique when preferences are strictly convex. In

ibed by just two parameters: the marginal tax rate and the

mp-sum subsidy.

The linear tax structure corresponds to

Under a linear tax system a household with ability s supplying l units of labor will pay tax of

tslslT +=

subsid ds.

sures that the budge sets of all households are

c

addition, the tax system is descr

lu

proposals fornegative income tax schemes, in which

all households below a given income level receive a subsidy from the tax system.

amount,

)( (28)

t is the marginal rate of tax and

egative.

-t) by , the consumption function of the household is

(29)

Ea

he first-order conditions can be reduced to

where is a lump-sum subsidy if positive and a tax if

n

Denoting (1

x =+sl.

ch household chooses consumption and labor supply to maximize utility (3.2) subject to (29).

T

=U

U

sl

x

. (30)

Labor suppl and functions cany and consumption dem be written as,

l s

x sl s

=

= +

( , , )

( , , )

l

(31)

ubstituting (31) into the utility function, there determine the indirect utility function,S



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(UU ) ),,(), sVs,(),,,( lssl= =+ (32)

with

slUV

UV

xx ==

,

(33)

where V is equal to the marginal utility of income.

The governments optimization problem is to choose the parameters of the tax system to

maximize social welfare subject to raising the required revenue,R.

(34)

[ ]( , , ) ( ) (35)

33) and defining the social marginal utility of income for a household of ability s by

)()),,((max dsssVw 0,subject to

+

( )10 sl = s s ds R

Using (

),,()),,

sVs (36) ((')( Vws =

The necessary conditions for the choice of and respectively are

( )s ds H =

00

( ) ( )

zs ds

1 (37)

and

=

0 0()( zdssz

)()1 dss

(38)

His th H s ds=

( )0 .

z

e population size,where



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Divide (38) by (37) and denote by a bar term of the formx/H.

z s ds

s ds

z( )

( )

0

0

0

0

z s ds

zs ds

( ) ( )

( ) ( )

1

1 1

=

(39)

on the left-han

welfare-weighted average labor supply. From totally differentiating the government revenue

onstraint whilst holdingR constant, it can be found that

The term d side of (39) is now denoted z() and can be interpreted as the

c

+

=0

)()1( dssz

zt

0)()1(1 dss

zRconst

(40)

(39) and (40),

Hence from

constRt

=)(

ince averaging over the p st give

z (41)

S opulation mu z z z( , )= , it follows from (41) that, holding

revenue constant

)(

zzzzzz

=+= (42)

Therefore (40) can be written in the form

RR constconst

constR

zt

z

=

(43)

ll that t= 1

zz

zz

= )()1()(

Reca



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const

z

zz

t

R

=

)((44)

here the derivative is taken with revenue constant.

Although the tax rule (44) only provides an implicit expression fort, it can be used to assess

fects of various parametric changes.

of s andz on increasing function

increased on the high-s households so that equity was given less weight.

3.

y into account.

he optimal structure of income taxation is characterized by applying Pontryagins maximum

principle.

The revenue function,

( )x s s ds) ( ' ) ( ' ) '= (45)

he level of utility u(s), pre-tax incomez(s) and the tax payments of households of ability s are

and the derivative of gross inco

w

the ef A reduction in the optimal tax would occur, with

a decreasing function of s, if the welfare weights were

9 Non-linear taxation

With non-linear taxation, the self-selection constraint must be taken full

T

R s z ss

s( ) ( '

1

2

T

me, ( )s z 'staken as the state variables is taken as the

ontrol variable. The level of consumption can then be found by solving

u

c

u s x s z s s( ( ), ( ), )=

Adopting a utilitarian objective, the control variable is chosen to mazimize

( ) .

u s s dss

s( ) ( )

1

2

(46)

subject to

( ) )()()( ssxszR

s (47)

R s R s( ) ( )1 2 0= = (48)

( )

u

s

u x s z s ss= ( ), ( ), (49)



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z

s

s= ( ) (50)

( )

z

ss

= ( ) 0 (51)

Introducing the adjoint variables (s), (s), (s) and(s), the Hamiltonian for the

The revenue constraint is captured by (47) and (48). To simplify, it is assumed that zero

revenue is to be collected. The rate of change in revenue (47) is derived directly from (48).

The self-selection constraint is represented by (49)(51); the first-order condition is (49), the

second-order condition are (50) and (51).

optimization is

( ) ( )H u s s s z s x s s s u x s z s ss= + +( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ), ( ),

v s s s s+ +( ) ( ) ( ) ( ). (52)

an ns ared the necessary conditio

H v= + =' ( ) 0 (53)

( )[ ] ( ) H z s x s sz

u x s z s ss( ) ( ) ( ) ( ), ( ),

z zv= + = ' (54)

( )[ ] ( )

u u

H

us

z s x s s u x s z s ss= +

+ = ( )( ) ( ) ( ) ( ), ( ),

' (55)

R

H= =' 0 (56)

z

s= 0,

h transversality con

0 (57)

wit ditions are



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( ) ( ) ,s s1 2= = ( ) ( )1 2v s v s0 0= = (58)

To derive the form of these conditions that will be used below, note that from the identity( )s z s s, ( ),

u x( ) it follows that

x

z

u

ux

z

= = (59)

and

x

u ux=

1. (60)

In addition

( )u x s z s ss ( ), (

zu

x

zu usx z x s

),= + = (61)

and

( )

x

sxsx u

u

uu s xu

sszsxu==

),(),((62)

( )Now denoting ( ) ( ) ' ( )s s s= , (53)(58) can be rewritten,

=

+ ux s ( ) '1 0 (63)

'usx + +

=u ux x

1 0 (64)

z

s= 0 0 (65)

( ) ( ) ,s s1 2= ( ) ( )s s0 0= =1 2 = (66)



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The interpretation of these neces

(1) is zero for allself-selection constraint is not binding and pre-tax income is a stirictly increasing

irst-order approach is identical to the second-order approach.

(2) is not zero for all s. If is positive over [s0, s1], all households with abilitiesfalling in this interval earn the same pre

These households are bunched at a single income level. Furthermore they must have the

utility is increasing with lds since those with highers have to work

less to obtain the co e.

There are several thoretical results on optimal income tax.

orem 1

sary conditions is as follows.

s. The second order condition for the satisfaction of the

function of ability. The f

-tax income.

same level of consumption. Note that although pre-tax income and consumption are identical,

s over the bunched househo

mmon level of incom

The (Mirrle

If there exists an ability level ss 0 such that l(s0)=0, then l(s)=0 for any s


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Theorem 4 (Seade (1982))

Let the upper bound on

household of ability s2.roof

ability s2 be finite. Then the marginal rate of tax must be 0 for a

P

les (1995, pp.151-

short, since there is no household beyond ability s2 , there is no point to set the marginal rate

ax positive at s2.

Th

See My 2).

In

of t

eorem 5 (Seade (1977))

For a population with bounded ability, any income tax schedule with a positive marginal

rate at the top of the scale can be replaced by one that leaves all households better off,

including them to earn more income but paying the same tax.

Proof

See Myles (1995, pp.153).

Theorem 6 (Seade (1977))

If there is no bunching at the lowest income, the optimal marginal rate for the household

of lowest ability is zero.

Proof

Se m 4.

been derived. arginal rate

1. must be

imal tax function cannot be progressive. In other

ords, it may be optimal to force some households to choose to undertake no work. In this

To generate numerical results, Mirrlees (1971) assumed that the social welfare function took

e Myles (1995, pp.154). This is a Mirror image of Theore

The above results of the optimal non-linear tax have The optimal m

of taxation must be between 0 and At the highest and lowest abilities, the tax rate

zero. The latter finding shows that the opt

w

case, it is the lowest ability households that will not work. Pre-tax income and consumption

must both be increasing functions of ability.

3.10 Numerical Results

the form



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0,

01

0

=

=

v

v>ev

w v

(67)

nt greater concern for equity, with v=0 representing the utilitarian

case.

The s the Cobb-Douglas,

)(

,)(

0=

dssU

dssU

Higher values of v represe

individual utility function wa

U x l= + log log( )1 (68)

og-normal,and the skill distribution is l

( )

+=

2

)1(exp)(

2s

ss (69)

ith a standard deviation (=0.39 from Lydall (1968)). An implicit assumption is that the

able 3.1 Optimal Tax Schedule

Consumption Average tax (%) Marginal tax (%)

log1

W

skill distribution can be inferred directly from an observed income distribution.

T

Income

(a)zG=0.013, v=0,=0.390.00 0.03 --- 230.05 0.07 -34 260.10 0.10 -5 240.20 0.18 9 210.30 0.26 13 190.40 0.34 14 18

0.43 15 160.50

(b)zG=0.003, v=1,=0.390.00 0.05 --- 30

0.08 -66 340.12 -34 32

0.34 16 2217 20

0.050.100.20 0.19 7 280.30 0.26 13 250.400.50 0.41

(c)zG=0.013, v=1,=10.00 0.10 ---0.10 0.15 -50

5058

0.25 0.20 20 600.50 0.30 40 591.00 0.52 48 571.50 0.73 51 542.00 0.97 51 523.00 1.47 51 49

Source: Mirrlees (1971)



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The most important feature of the first two panel (a) and (b) in Table xx is the low marginal

rates of tax, with the maximal rate being only 34%. There is also limited deviation in thesetes. The marginal rates become lower at high incomes but do not reach 0 because the skill

verage rate of tax is negative for low incomes so that

w-income consumers are receiving an income supplement from the government.

eviation from 0.39 to 1.00). This raises the marginal tax rates but there remain fairly

onstant across the income range. Kanbur and Tuomala (1994) find that an increased

dispersion of skills raises the marginal tax rate at each income level and that it also has the

ct of moving the maximum tax

increasing over the majority of households.

Atkinson (1975) considered the effect of changing the social welfare function to the extreme

axi-min form,

in{U} (70)

From the above table, it can be seen that increased concern for equity, v going from 0 to 1,

reased the optimal

onsiderations such as maxi-min SWF lead to high marginal rates?. The result is given the

ads to generally higher taxes. However they are again

highest at low incomes and then decline. Absolute rate is lower than expected in all cases.

Table 3.2 Optimal tax Schedule: Utilitarian vs. Maxi-min

Utilitarian Maxi-min

ra

distribution is unbounded. The a

lo

The panel (c) of Table 3.1 show the effect of increasing the dispersion of skills (changing

standard d

c

effe rate up the income range, so that the marginal tax rate is

m

w=m

inc marginal tax rates. The natural question is can strong equity

c

below table. The maxi-min criterion le

Level ofs Average rate (%) Marginal rate (%) Average rate (%) Marginal rate (%)Median 6 21 10 52Top decile 14 20 28 34Top percentile 16 17 28 26

Source: Atkinson and Stiglitz (1980, Table 13-3, p.421)

3.11 Numerical Results

To generate numerical results, Mirrlees (1971) assumed that the social welfare function took

the form

6

6 This part draws from Myles (1955) Chap 2, pp.156-9.



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0,

,)(0

=

=

v

v>dssw (71)

Higher values of v represent greater concern for equity, with v=0 representing the utilitarian

case.

y function was the Cobb-Douglas,

)( =

dss

01 vU

0 U

ev

The ind al utilitividu

U x l= + log log( )1 (72)

and the skill distribution is log-normal,

( )

+=

2

)1log(exp

1)(

2s

ss (73)

With a standard deviation (=0.39 from Lydall (1968)). An implicit assumption is that the

skill distribution can be inferred directly from an observed income distribution.

Table 3 Optimal Tax Schedule

Income Consumption Average tax (%) Marginal tax (%)

(a)zG=0.013, v=0,=0.390.000.05 0.07

0.03 --- 23-34 26

0.10 0.10 -5 24

0.40 0.34 14 180.50 0.43 15 16

0.20 0.18 9 210.30 0.26 13 19

(b)zG=0.003, v=1,=0.390.00 0.05 --- 300.05 0.08 -66 34

0.10 0.12 -34 320.20 0.19 7 280.30 0.26 13 250.40 0.34 16 220.50 0.41 17 20

(c)zG=0.013, v=1,=10.00 0.10 --- 50

580.25 60

4048

1.50 0.73 51 540.97 51 521.47 51 49

0.10 0.15 -50200.20

0.300.501.00 0.52

5957

2.003.00

Source: Mirrlees (1971)



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The most important feature of the first two panels (a) and (b) in Table 3 is the low marginal

tes of tax, with the maximal rate being only 34%. There is also limited deviation in these

n is unbounded. The average rate of tax is negative for low incomes so that

w-income consumers are receiving an income supplement from the government.

The panel (c) of Table 3 show the effect of increasing the dispersion of skills (changing

dard deviation from 0.39 to 1.00

constant across the income range. Kanbur and Tuomala (1994) find that an increased

dispersion of skills raises the marginal tax rate at each income level and that it also has the

ffect of moving the maximum tax rate up the income range, so that the marginal tax rate is

inson (1975) considered the effect of changing the social welfare function to the extreme

m

=min{U}

rom the above table, it can be seen that increased concern for equity, v going from 0 to 1,

increased the optimal marginal tax rates. The natural question is can strong equity

onsiderations such as maxi-min SWF l

below table. The maxi-min criterion leads to generally higher taxes. However they are again

ighest at low incomes and then decline. Absolute rate is lower than expected in all cases.

Utilitarian Maxi-min

rarates. The marginal rates become lower at high incomes but do not reach 0 because the skill

distributio

lo

stan ). This raises the marginal tax rates but there remain fairly

e

increasing over the majority of households.

Atk

axi-min form,

w (74)

F

c ead to high marginal rates?. The result is given the

h

Table 4 Optimal tax Schedule: Utilitarian vs. Maxi-min

Level ofs Average rate (%) Marginal rate (%) Average rate (%) Marginal rate (%)Median (50%) 6 21 10 52Top deci 20 28

17 28

litz (1980, T 3-3, p.421)

le (10%) 1416

3426Top percentile (1%)

Source: A on and Stigtkins able 1

3.12 Vo over a Flat T

Having tified the properties of the optimal tax structure, we now consider the tax system

that emer om o his, we consider people voting over tax

schedules me degree of redistribution. Bec it is difficult to model voting on

nonlinear tax schemes given the high dimensionality of the problem, we will restrict attention to

a linear tax structure as originally proposed by Romer (1975). We specify the model further

sary complications and to simplify the analysis of

ting ax

iden

ges fr the political process. T do t

that have so ause

with quasi-linear preferences to avoid unneces



32/43

the voting equilibrium.

, that individuals differ onl in their level of skill. We assume that skills

are distributed in the population according to a cumulative distribution function )s that isknown to everyone, with mean skill

Assume, as before y

(Fs and median Individuals work a consume.

They also vote on a linear tax scheme that pays a lu m benefit b to each individual

financed proportional inco ax at rate t. dual utility function has the

q

ms

m

The indivi

. nd

psu

by a me t

uasi-linear form

2

2

1,

=

s

zx

s

zxu , (75)

and the individual budget constraint is

bztx += ]1[ . (76)

It is easy to verify that in this simple model the optimal income choice of a consumer with skill

level s is

2]1[)( stsz = . (77)

preferences imply that there is no income effect on labor supply (i.e., )(sz is

independent of the lump-sum benefit b ). This simplifies the expression of the tax distortion

induces taxpayers to work less and earn less income.

rive

e second equality. This constraint says that the lump-sum benefit paid to each individual

termed the

Dupuit-Laf and d n of le

m uit-Laffer curve ell-shaped k a 2/1

The quasi-linear

and makes the analysis of the voting equilibrium easier. Less surprisingly a higher tax rate

The lump-sum transfer b is constrained by the government budget balance condition

)(]1[))(( 2sEttsztEb == , (78)

where )(E is the mathematical expectation, and we used the optimal income choice to de

th

must be equal to the expected tax payment ))(( sztE . This expression is

fer curve escribes tax revenue as a functio the tax rate. In this simp

odel the Dup is b w a peaith t =t d no tax collected at the

e 1= can now derive individ al preferences over tax schedules by

s angement (indirect) utility can be written

an

nds 0=t and t . We u

ubstituting (49) and (50) into (48). After re-arr



33/43

22]1[1

),,( stbsbtv += .2

(79)

Taking the total differential of (52) gives

dtstdbdv 2]1[ = . (80)

so that along an indifference curve where 0=dv ,

2]1[ stdt

db= . (81)

It can be seen from this that for given t, the indifference curve beco es steeper in ),( bt

space as

m

s increases. This monotonicity is a consequence of the single-crossing property of

the indifference curves. The single-crossing property is a sufficient condition for the Median

ter Theorem to apply. It follows that there is only one tax policy that can result from

jority voting: it is the policy preferred by the median voter (half the voters are poorer than the

dian and prefer hig

Letting mt he tax rate preferred by the median voter, then we have mt implicitly defined

the solution to the first-order condition for maximizing the median voters utility. We

into account the government budget constraint (51)

obtain

Vo

ma

y

me her tax rates, and the other half are richer and prefer lower tax rates).

be t

b

differentiate (52) with respect to t, taking

to

22 ]1[)(]2 stsEtv

=

. (82)

etting this expression equal to zero for the median skill level ms yields the tax rate preferred

by the median voter

1[t

S

22 )(2 mm

ssEt

= , (8

or, using the optimal income choice (50),

22 )( mssE 3)

m

m

zzE

)(2

)mzzE

t

=)(

. (84



34/43

This simple model predicts that the political

f the median voter in the income distribution. The greater is income inequality as measured

that the

ses

mpirical Fact in Japan

ble 5 Individual Inhabitants

Inhabitants Tax Income Tax

equilibrium tax rate is determined by the position

oby the distance between median and mean income, the higher the tax rate. If the median voter

is relatively worse off, with income well below the mean income, then equilibrium

redistribution is large. In practice, the income distribution has a median income below the

mean income, so a majority of voters would favor redistribution through proportional income

taxation. More general utility functions would also predict extent of this redistribution

decrea with the elasticity of labor supply.

E

Ta and Income Tax

Tax ecipient Municipal governments on 1st January National governmentR

Tax government Individual lives in JapanPayer Individual lives in a municipal

Tax ethod With holding AssessmentM

Tax ase Last years income This years incomeB

Income Deduction

Basic deduction

Spou

Special spouse d

Family deduction

330,000

330,000

Basic deduction

Family deduction

380,000

380,000

se deduction

eduction

330,000max 330,000

Spouse deduction

Special spouse deduction

380,000max 380,000

MiniInc

A couple with 2 children

3,031,000,373,000

A couple with 2 children

Before a special tax deduction 3,616,000After special tax deduction 4,917,000

mum Taxableome (1998) Before a special tax deductionAfter special tax deduction 4

Taxable Income Tax Rates Taxable Income Tax RatesPrefectural Municipal Total

Above 7,000,000 3% 12% 15% Below 18,000,000Below 30,000,000

30%40%50%

Below 2,000,000Below 7,000,000

2%2%

3%7%

5%10%

Below 3,300,000Below 9,000,000

10%20%

Above 30,000,000Tax Rate Capital Gain Income Capital Gain Income

Below 40,000,000Below 80,000,000Above 80,000,000

2%2%3%

4%5.5%

6%

6%7.5%

9%

Below 40,000,000Below 80,000,000Above 80,000,000

20%25%30%

In 1999-2001 In 1998-2000

Below 60,000,000Above 60,000,000

2%2%

4%5.5%

6%7.5%

Below 60,000,000Above 60,000,000

20%25%

Prefectural 1,000MunicipalPopulation above 0.5 millionPopulation between 0.5 millionPopulation below 50,000

3,0002,5002,000

Tax- Tax Incentives

Housing Acquisition

Research & Development

Exemption

- Adjustment for Double Taxation

Dividends

Foreign Tax

- Adjustment for Double Taxation

Dividends

Foreign Tax

Tax Revenue (1996) 90,17 trillion yen 189,649 trillion yen



35/43

Ta Tax Rates Schedule

1987 1988 1989 1995 1999

ble 6 Historical Changes in Income

1974 1984

% % % % % million % million % million

Tax R4045

4550ate

10

212427303438424650

60657075

10.5

1721253035

50556065

10.5

2025303540

5560

10

405060

10

4050

(~3)

(~2)(2~)

10

4050

(~3.3)

(~30)(30~)

10

37

(~3.3)

18)(18~)

M mumInhabitants

Tax Rate18% 18% 18% 16% 15% 15% 13%

12141618

1214

1216

2030

2030

(~6)(~1)

2030

(~9)(~18)

2030

(~9)(~

55 70

axi

CombinedMaximum 93% 88% 78% 76% 65% 65% 50%Tax Rate

NT

umber ofax Brakets

19(13)

15(14)

12(14)

6(7)

5(3)

5(3)

4(3)

M

inimumTaxableIncome

1,707,000 2,357,000 2,615,000 2,619,000 3,198,000

3,539,000(3,616,000

after the 1998amendment)

3,821,000

Table 7 Change in Income Tax Payers

Total

of Declared EarnersApplicantsfor Refund

SalaryEarners

DeclaredEarners

Business AgricultureSmall

BusinessOthers

1985 3,665 737 4,402 228 31 67 411 599

1986 3,728 770 4,498 231 32 70 437 654

,538 235 25 70 441 699

,689 245 24 70 441 696

1989 3,961 796 4,757 242 23 67 464 659

25 67 513 663

24 68 699

1992 4,403 858 47 24 69 735

4,473 22 781

4,478 27 867

4,484 5,286 213 19 510 864

4,537 213 20 531 883

1997 4,627 5,454 206 545 9

,551 2 557 ---

1987 3,767 771 4

1988 3,909 780 4

1990 4,158 855 5,013 250

1991 4,333 856 5,189 252 512

5185,261 2

1993 843 5,316 230 67 524

1994 822 5,300 223 62 510

1995 802 60

1996 824 5,361 60

827 16 60 09

1998 4,703 848 5 10 20 61

Source: Ministry of Finance.



36/43

Table 8 The 19 ome Bu R y

(million yen)

InAverageIncome

Average Average

xableInco

Average

Calculatedx

A e Taxnt

Gro

IncomRa

Effective

Income TaxRate

(-)

96 Declared Inc Tax rden ate b Income Class

come classIncome Ta

Deduction

me

Ta

verage Tax AveragDeduction Payme

ss

e Taxte

100 748 2 18 2 8.9%546 202 0 0 .4%100 200 1,540 56 48 3.1% 8.8%200 300 2,476 1,016 103 88 3 8.7%300 500 3,870 2,074 2 197 5 9.5%500 1000 6,899 4,879 635 9.2% 13.0%1000+ 23,005 893 21,112 32 4,707 20.5% 22.3%

TotalAverage

5,864 8 759 12.9% 17.7%

995 545 01,459 1 .5%1,796

20330

614

.1%2,0 70

4,7891,

1,572 4,292 795

Source: The 1996 Sample Su (Tax Bu

Earners, Total Salary and Ta

Number of Salary Earners (thousands) Total Salar Tax paid EffectiveTax Rate

rvey of Declared Income Tax reau).

Table 9 Number of Salary x in 1996

y (100 million yen)

Income Class of Tax Payers 00million yen)of Tax Payers (1

Share Share Share ShareShare

(%)- 100 3,228 7.2 479 1.2 23,993 1.2 2,789 0.1 213 0.2 7.6100 54,310 2.8 1,345 1.3 2.5200 4,818 10.7 3,419 8.7 72,423 3.5200 58 8.0 5,021 3.2300 6,818 15.2 6,230 15.9 173,522 8.4 1 ,778 4.9300 400 7,780 18 3.317.3 7,328 .7 272,122 13.2 25 13.06,351 8,471 8.2400 500 6,530 5 15.9 ,908 14 4 3.314. 6,244 292 .2 280,175 1 .2 9,222 9.0500 600 4,964 11.1 12. 272,721 ,861 8 3.44,802 3 13.2 263 13.4 8,898 .7600 700 3,273 7.3 3,215 8.2 211,900 ,179 7 7.2 3.610.2 208 10.6 ,429700 800 2,384 5.3 2,372 6.1 178,002 8.6 ,094 7 7. 4.4177 9.0 ,707 5

800 900 1,604 3.6 1,604 4.1 135,837 6.6 ,837 7 7. 5.3135 6.9 ,168 0900 1000 1,004 2.2 1,004 2.6 95,168 4.6 ,168 6 6.0 6.495 4.8 ,1371000 1500 1,963 4.4 1,963 5.0 232,610 1.2 232,610 11.8 21,461 20.9 9.211500 2000 378 0.8 1.0 64,247 3.1 64,247 3.3 9,475 9.2 14.73782000 2500 87 0.2 0.2 20,098 1.0 20,098 1.0 3,956 3.8 19.7872500+ 64 0.1 0.2 23,254 1.1 23,254 1.2 6,294 6.1 27.164

T 44,896 100.0 39,189 100.0 ,068,805 100.0 1,972,750 100.0 102,797 100.0 5.2otal 2

Note: Employees on December 31, 1996.

Source: The 1996 Tax Bur Private Sector Income Survey (Tax Bureau).

Table 10 Tax asticity

rect Tax (Income Tax) Total Tax

eau

El

DiIndividual Corporate Indirect Tax Revenue

1985 1.0 4 2 860 1.31 0.6 0.4 0.

1986 1.8 6 3 879 1.74 1.6 1.8 1.

1987 3.64 2.58 4.78 1.94 3.20

1988 2.2 8 4 925 1.57 2.8 1.0 1.

1989 1.6 2.736 0.58 0.77 1.44

1990 1.36 2 16 1.06.69 0. 0.06

1991 0.0 3 .10 0.032 0.06 1.4 0

1992 6.18 6.09 7.77 1.09 4.50

1993 4.14 2.14 17.71 4.71 1.86

1994 4.75 5.50 8.25 8.25 5.57

1995 1.10 2.00 6.50 1.55 1.20



37/43

Figure 6 Before Tax Household Income Distribution in Japan

Fraction

j15000 1.8e+07

0

.428925

Source : The National Survey of Family Income and Expenditure 1984, 1989 and 1994 (pooled)

Note: Monthly average income during September through November.

Figure 7 Before Tax Household Income Distribution in Japan (log normal transformation)

Fraction

j15000 1.8e+07

0

.125923

Source : The National Survey of Family Income and Expenditure 1984, 1989 and 1994 (pooled)

Note: Monthly average income during September through November.



38/43

Figure 8 After Tax Household Income Distribution in Japan

Fraction

ap2697 1.4e+06

0

.08466

Source : The National Survey of F Incom d Ex e 1984, 1989 a 4 (po

: Mon rage e d eptem throu mber.

re 9 r T ou In e D ion ap no transformatio

amily e an penditur nd 199 oled)

Note thly ave incom uring S ber gh Nove

Figu Afte ax H sehold com istribut in J an (log rmal n)

Fraction

ap2697 1.4e+06

0

.123219

Source National Survey mily Income xpenditure 1984, 1989 and 1994 (pooled)

Note: Monthly average incom ng Septemb ugh Novemb

: The of Fa and E

e duri er thro er.



39/43

Exercises

1.

[Atkinson and Stiglitz (1980), p.376]

For the utility function vLX

Ai

in

ii

i

=

=

11

11

1

, where L units of labor, wage is w thus the

budget co household is

U

nstraint for the == ywLXq ii . Show that the income termsyxi / ms are zero. Derive the optimal tax structure where i( ) and cross price ter are

(positive) constants.

2. Recently many countries adopt indirect taxation (e.g. VAT) and shift its weight from directtaxation (e.g. individual income). Could you justify this shift of tax reform?

3. Stiglitz once argued that it can be shown, that if one has a well-designed income tax,adding differential commodity taxation is likely to add little, if anything. Would you

agree with him? Or in what circumstances does the use of commodity taxation allow a

higher level of social welfare to be achieved lwtbx ]1[ += in the presence of income

1]

Consider the bud lwtbx ]1[

taxation?

4. [Hindriks and Myles (2006) Chapter 15, Exercises 15.get constraint = + . Provide an interpretation of b. How

How is the choice of fected by increases in b andt? effects.

5. [Hindriks and Myles (2006) Chapter 15, Exercises 15.2]Assume that a consumer has preferences over consumption and leisure described by

]l=U , where

does the average rate of atx change with income? Let utility be given by 2l=xU .

l af Explain these

x1[x is consumption and l is labor. For a given wage rate w , which

her labor supply: an income tax at constant rate t or a lump-sum taxleads to a hig T that

raises the same revenue as the income tax?

6. [Hindriks and Myles (2006) Chapter 15, Exercises 15.3]Let the utilit ly function be = )log(xU . Find the level of labor supply if the wage rate,

w, is equal to 10. the effect of the introduction of an overtime premium that raises

w o s in excess of that worked at the wage of 10?7. [Hindriks and Myles (2006) Chapter 15, Exercises 15.4]

Assu )1log()log( l

What is

r hourto 12 f

=Ume that utility is x . Calculate the labor supply function.

Explain the form of this function by calculating the income and substitution effects of a

6]

Show that a tax function is average-rate progressive (the average rate of tax rises with

AR

wage increase.

8. [Hindriks and Myles (2006) Chapter 15, Exercises 15.TMRTincome) if > .

9.

[Hindriks and Myles (2006) Chapter 15, Exercises 15.7]



40/43

Which is better: a uniform tax on consumption or a uniform tax on income?

10. [Hindriks and Myles (2006) Chapter 15, Exercises 15.8]Consider the utilit

2

l=xU .a. For 10=U , plot the indifference curve with l on the horizontal axis andy function x on the

vertical axis.

Now define ls= . For 5.0b. z =s , 1, and 2 plot the indifference curves for 10=U withz on the horizontal axis andx on the vertical.

c. Plot the indiff curves for 5.0erence =s 1, and 2 thro e 20=x , 2=ugh th point z .d. any point (x zProve that at , ) the indifference curve of a high-skill consumer is flatter

than that of a low-skill.

11. [Hindriks and Myles (2006) Chapter 15, Exercises 15.9]Consider an economy with two co 11 =nsumers who have skill levels s and 22 =s and

22/110 l= xU . Let the governm come tax function that

4= , 5

utility function

leads to the allocation

ent employ an in

x =z for the consumer of skill 1=s and 9=x , 8z for

e compatibility constraint that each

consumer must prefer his allocation to that of the other.

high-skill to the low-skill consumer.

i. Calculate the effect on each consumers utility.ii. Show that the sum of utilities increases.iii. Show that the incentive compatibility constraint is still satisfied.iv. Use parts i through iii to prove that the initial allocation is not optimal for a utilitarian

social welfare function.

12. [Hindriks and Myles (2006) Chapter 15, Exercises 15.11]Assume that skill is uniformly distributed between 0 and 1 and that total population size is

normalized at 1. If utility is gi )1 l

=

the consumer of skill 2=s .

a. Show that this allocation satisfies the incentivb. Keeping incomes fixed, consider a transfer of 0.01 units of consumption from the

ven by log()log( = xU

al values ofb andtwhe

and the budget constraint is

ls , find the optim n zero revenue is to be raised. Isthe optimal tax system progressive?

13. [Hindriks and Myles (2006) Chapter 15, Exercises 15.13]12 s> . Denote

at to the high-skill consumer by

2x , 2z .

a.

tbx )1( +=

Consider an economy with two consumers of skill levels 1s and 2s , s

the allocation to the low-skill consumer by 1x , 1z and th

For the utility function szxuU = )( show that incentive compatibility requires that

)]1 .()([ 21 xuxuz +2z =



41/43

b. For the utilitarian social welfare function2

22 )( s

zxu , express W as

)( hx , derive the

itution for th

alloc

14. (2006) Chapter 15, Exercises 15.15]revenue is to

]1,0[

1

11)( s

zxuW +=

a function of 1x and 2x alone.c. Assuming log)( hxu = optimal values of 1x and 2x and hence of 1z d. Calculate the marginal rate of subst e two consumers at the optimal

ation. Comment on your results.

[Hindriks and Myles

and 2z .

Suppose two types of consumers with skill levels 10 and 20. There is an equal number of

consumers of both types. If the social welfare function is utilitarian and no

be raised, find the optimal allocation under a nonlinear income tax for the utility function

l= )log(xU . Contrast this to the optimal allocation if skill was publicly observable.

15. [Hindriks and Myles (2006) Chapter 15, Exercises 15.16]Tax revenue is given by )()( ttBtR = , where t is the tax rate and )(tB is the tax

base. Suppose htat the tax elasticity of the tax base ist

t

=

1with

1,

2

1 .

ng tax rate?a. What is the revenue-maximizib. Graph tax revenue as a function of the tax rate both for 2/1= and 1= . Discus

= 1 lU

s

the implications of this Dupuit-Laffer curve.

16. [Hindriks and Myles (2006) Chapter 15, Exercises 15.17]Consider an economy populated by a large number of workers with utility function

1] , where[x x is disposable i come, l is the fraction of time worked

( 10 l ), and

n

is a preference parameter (with 10 B is

yment.

gh-skill person have higher

that the condition for job market

g

the unconditional benefit pa

a. Find the optimal labor supply for someone with ability w . Will the high-skill personwork more than the low-skill person? Will the hi

disposable income than the low-skill person? Show

participation ist

w

>1

[[ B ]]1 .

ds are only used to fin e the benefitb. If tax procee Banc , what is the governments budgetconstraint?

c. Suppose that the mean skill in the population is w and that the lowest skill is afraction of the mean skill. If the government wants to redistribute all tax1


42/43

),( Bt satisfy?

d. Find the optimal tax rate if the government seeks to maximize the disposable income ofbject to evethe lowest skill worker su ryone working.



43/43

xation with General Equilibrium Effects on Wages,

blic Economicstiglitz

ublic Economics, 6, 55-75.

iversity Press.ics, McGraw Hill

Boa

Crem

Devereux, M.P. (19

gressivity: A nt,Na (1),Hillm bridge University Press.Hin omics, T Press.Kanbur, S.M.R. and M. Tuomala (1994) Inherent, Inequality and the Optimal Graduation of

f Economics, 96, 275-82.

iversity Press.

r Developing Countries, Oxford

RomTax,Journal of Public Economics, 7, pp.163-8.

Rosen, H.S. (1999) PSeade, J. (1977) On the Shape of Optimal Tax Schedule, Journal of Public Economics, 7,

203-36Sea e

Stern, N. n the Specificat dels of

Tuo x and Redistribution, Oxford University Press.

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Allen, F. (1982) Optimal Linear Income Ta

Journal of Pu , 6, 123-62.Atkinson, A. and S , J. (1976) The design of tax structure: direct vs. indirect taxes,

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Economic Studies, 38, 175-208.Myles, G.D. (1995) Public Economics, Cambridge Un

Newbery, D. and N. Stern (1987) The Theory of Taxation foUniversity Press.

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.de, J. (1987) On the Sign of the Optimum Marginal Incom Tax, Review of Economic

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